The temperature of a chemical solution is originally $21^\circ\text{C}$. A chemist heats the solution at a constant rate, and the temperature of the solution is $75^\circ\text{C}$ after $12$ minutes of heating. The temperature, $T$, of the solution in $^\circ\text{C}$ is a function of $x$, the heating time in minutes. Write the function's formula. $T=$
Explanation: The chemist heats the solution at a constant rate, so we're dealing with a linear relationship. We could write the desired formula in slope-intercept form: $T= mx+ b$. In this form, $ m$ gives us the slope of the graph of the function and $ b$ gives us the $y$ -intercept. Our goal is to find the values of $ m$ and $ b$ and substitute them into this formula. We know that the initial temperature of the solution is $21^\circ\text{C}$, so the $y$ -intercept ${b}$ is ${21}$, and our function looks like $T={m}x+{21}$. We also know that after $12$ minutes of heating, the temperature is $75^\circ\text{C}$, which means when $x=12$, $T=75$. We can use this and the $y$ -intercept to find ${m}$ : $\begin{aligned} {m}&=\dfrac{T_2-T_1}{x_2-x_1} \\\\ &=\dfrac{75-21}{12-0} \\\\ &=\dfrac{54}{12} \\\\ &=\dfrac{9\cdot\cancel{6}}{2\cdot\cancel{6}} \\\\ &={4.5} \end{aligned}$ This means the temperature of the solution increases by $4.5^\circ\text{C}$ per minute. Since ${m}={4.5}$ and ${b}={21}$, the desired formula is: $T={4.5} x + {21}$